MATH461 RINGS AND MODULES
| Course Code: | 2360461 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 6.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Undergraduate |
| Course Coordinator: | Assoc.Prof.Dr. SEMRA ÖZTÜRK |
| Offered Semester: | Fall Semesters. |
Course Objectives
This course is to provide the background for students who are willing to learn more about rings and modules which are the fundamental mathematical structures occuring everywhere ! It is useful for everyone but especially for students who are planning to study any algebra related topics such as algebraic topology, algebraic geometry, analysis.
Approximately half of the semester will be on rings, the second half will be on modules. Rings will be a more detailed but much faster version of some of the topics you have seen in Math 367, and Math 116. Modules will be new to you. They are generalizations of vector spaces also generalization of abelian groups. (Modules over group algebras|rings are examples of groups acting on vector spaces.)
Thus in module theory linear algebra comes up quite often. You should be comfortable using linear algebra to get more out of this course. We will see the primary decomposition theorem for finitely generated modules over a Euclidean domain. Usually there is not enough time to cover tensor products.
Course Content
Classical theory of rings, ideal theory, isomorphism theorems. The group ring. Localization. Submodules, direct products direct sums, factor modules and factor rings. Homomorphisms. Classical isomorphism the-orems. The endomorphism ring of a module. Free modules, free and divisible abelian groups. Tensor product of modules. Finitely generated modules over principal ideal domains.
Course Learning Outcomes
At the end of this course a student should acquire
- A solid understanding of the basic concepts and definitions related to rings and modules
- The Structure Theorem for finitely generated modules over e Euclidean domain
- Tensor products (if time permits)
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Can produce innovative thoughts and products. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Has a comprehension of mathematical symbols, concepts together with the interactions among them and can express his/her solutions similarly. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Is responsive to life-long learning, improving his/her skills and abilities | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution